534 



NATURE 



[January 14, 191 5 



the same track, and the author refers to Sir 

 John Leslie, Canon Moseley, Prof. Goodsir (on 

 whose tomb in Edinburg-h a log-arithmic spiral was 

 graven), and Mr. A. H. Church, whose studies 

 of phyllotaxis are well known. It is a track to 

 be followed. A symbol, which adorns the cover of 

 the book before us, is held by the author to indicate 

 that Chinese philosophy had adopted the log- 

 arithmic spiral as a symbol of growth so long ago 

 as the twelfth century, and several investigators 



{Country Life. 



Fig. I. - Central shaft o' spiral staircase in the wing of Francois I., 

 Chateau of Blois. From " The Curves of Life." 



are still engaged in the twentieth century in the 

 search for that symbol (which we now call a 

 formula). Nor can we forget that Leonardo da 

 Vinci, whose manuscripts the author has studied, 

 was greatly interested in spiral shells — of both 

 living and extinct types, for he was one of the 

 Hrst to appreciate fossils — and used them in his 

 art. Mr. Cook brings forward evidence that the 

 spirals in natural forms m.ade a deep impression 

 on the early artists ; and the development of this 



NO. 2359, VOL. 94] 



idea forms a very interesting part of the book, 

 the parallelism worked out between shells and 

 staircases being especially skilful. 



Spirals are sometimes seen in the inorganic 

 domain — in whirling dust, in waterspouts, in starry 

 nebulae, and in certain crystals. But it is in the 

 organic realm that they abound, and Mr. Cook's 

 record of them is impressive. In Foraminifer and 

 Nautilus, Venus Flower Basket and the horns of 

 an antelope, the typical spermatozoon and the 

 egg-capsule of the Port Jackson shark, the um- 

 bilical cord and the cochlea of the ear, the arms 

 of a Lamp-shell and the trachea of an insect, the 

 fir-cone and the Turbine shell, the antherozooid 

 of a Cycad and the stem of the honeysuckle, the 

 spiral vessel of a melon and the twisted fibrils of 

 the muscle-fibres of some molluscs, the fruit-stalk 

 of Cyclamen and the inflorescence of Forget-me- 

 not, the leaves on the stems which find in a 

 Fibonacci series the minimum superposition and 

 the maximum exposure, the seaweed Vidalia like 

 a curled shaving and the still more beautiful stair- 

 case of the rare Riella helicopJiylla from Algiers, 

 the twisting tendrils of the vine 

 and the horns of a ram, the 

 double spiral of osseous lamellae 

 in the shaft of long bones and 

 the staircase within the skate's 

 intestine, a Spirochaete and 

 the muscle-fibres of an arteri- 

 ole ! Sometimes their utility 

 is plain, e.g., in assisting in 

 the distribution of seeds and 

 sperms; or in acting like a 

 spring, as in tendrils and 

 tracheae ; or in allowing many 

 structures to be crowded in a 

 small space, as in buds ; or in 

 giving architectural strength 

 as in the lamellae of bones. 

 Mr. Cook points out shrewdly 

 that there is very often a suggestion of growth 

 under resistance, which might be illustrated by 

 the young leaves in a bud, or the spiral fibre of 

 insect tracheae, or the Venus Flower Basket build- 

 ing itself up on the floor of the Deep Sea ; and 

 we like Mr. Schooling's suggestion that the cf> 

 proportion is an expression of economy of form. 



There is no doubt that Goodsir had a definite 

 expectation that some physiological law of growth 

 o:' "production " might be discovered by a further 

 study of the recurrent spiral form. We are in- 

 debted to Mr. Cook for re-directing attention to 

 this quest. 



There .are, of course, numberless logarithmic 

 spirals, and Mr. Cook, with assistance from Mr. 

 Mark Barr and Mr. William Schooling, has deter- 

 mined that the form which is most descriptive 

 of those in organic nature is one which he calls 

 the (f) spiral — one in which the ratio of increase 

 (<^) is (i±is/^)^2, or i'6i8o34, the sum of any 

 two terms being equal to the next term. This 

 kind of spiral is illustrated approximately by 

 many very successful forms of organic growth. 

 We must emphasise the word approximately, for 



Fig. 2. — The common- 

 form of Valuta vesper- 

 tilis (section). From 

 " The Curves of Life." 



